If $(\sec A + \tan A)(\sec B + \tan B)(\sec C + \tan C) = (\sec A - \tan A)(\sec B - \tan B)(\sec C - \tan C)$,then each side is equal to

Let $f_k(x) = \frac{1}{k}(\sin^k x + \cos^k x)$,where $x \in R$ and $k \ge 1$. Then $f_4(x) - f_6(x)$ is equal to

The number of real solutions of the equation $(\sin x - x)(\cos x - x^2) = 0$ is

It is known that $\sin \beta = \frac{4}{5}$ and $0 < \beta < \pi$. Then the value of $\frac{\sqrt{3} \sin(\alpha + \beta) - \frac{2}{\cos(\pi/6)} \cos(\alpha + \beta)}{\sin \alpha}$ is:

If $\frac{\tan (\alpha+\beta-\gamma)}{\tan (\alpha-\beta+\gamma)}=\frac{\tan \gamma}{\tan \beta}$ and $\beta \neq \gamma$,then the value of $\sin 2 \alpha+\sin 2 \beta+\sin 2 \gamma$ is

If $\sin (y+z-x), \sin (z+x-y)$ and $\sin (x+y-z)$ are in $A$.$P$.,then